تقرير
The Second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data
| العنوان: | The Second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data |
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| المؤلفون: | Burq, Nicolas, Camps, Nicolas, Latocca, Mickaël, Sun, Chenmin, Tzvetkov, Nikolay |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, 35Q55, 35A01, 35R01, 35R60, 37KXX |
| الوصف: | We consider the Wick ordered cubic Schr\"odinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e. a stable closed geodesic), which prevents any regularization in the second Picard iteration. Comment: Minor changes |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2404.18241 |
| رقم الأكسشن: | edsarx.2404.18241 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |